Normal families and shared values of meromorphic functions

Abstract

Let $\cal{F}$ be a family of meromorphic functions in a domain $D$, let $q, k$ be two positive integers, and let $a, b$ be two non-zero complex numbers. If, for each $f \in \cal {F}$, the zeros of $f $ have multiplicity at least $ k+1$, and $f=a \Leftrightarrow G(f)=b$, where $G(f)=P(f^{(k)})+H(f)$ be a differential polynomial of $f$ satisfying $q \geq \gamma_H$, and $\frac{\Gamma}{\gamma} |_H < k+1$, then $\cal {F}$ is normal in $D$.